Relative permeability models

Relative permeability is probably one of the least-defined parameters in chemical flooding processes. The classical relative permeability curves represent a situation in which the fluid distribution in the system is controlled by capillary forces. When capillary forces become small compared to viscous forces, the whole concept of relative permeability becomes weak. This area has not been adequately researched, and theoretical understanding is rather inadequate (Brij Maini, University of Calgary in Canada, personal communication, 2007). This section discusses relative permeability models related to surfactant flooding and the IFT effect on relative permeabilities. 

M. Delshad, and G. A. Pope, Comparison of the Three-Phase Oil Relative Permeability Models, Transp. Porous Media, (4) 59-83,1989. 

Lloyd and Dodds [1972] calculated the flow of liquid firomthe cake by application of Darcy s law to elemental sections of the cake using a relative permeability model Wakeman [1979] extended this approach by incorporating a pore distribidion function in the relationship between saturation and capillary pressure. Ihe model involves a con q)utation of liquid and gas flow by a simultaneous solutions of Darcy s law. The relative permeability values used are obtained from a pore size disbibotian index X which relates the average reduced saturation of the cake to the ratio of the modified thrediold pressure Pb to the capillary pressure Pc. The equations are ... 

Conventionally, the sample is initially saturated with one fluid phase, perhaps including the other phase at the irreducible saturation. The second fluid phase is injected at a constant flow rate. The pressure drop and cumulative production are measured. A relatively high flow velocity is used to try to negate capillary pressure effects, so as to simplify the associated estimation problem. However, as relative permeability functions depend on capillary number, these functions should be determined under the conditions characteristic of reservoir or aquifer conditions [33]. Under these conditions, capillary pressure effects are important, and should be included within the mathematical model of the experiment used to obtain property estimates. 

The overall gain of the multiphase mixture model approach above is that the two-phase flow is still considered, but the simulations have only to solve pseudo-one-phase equations. Problems can arise if the equations are not averaged correctly. Also, the pseudo-one-phase treatment may not allow for pore-size distribution and mixed wettability effects to be considered. Furthermore, the multiphase mixture model predicts much lower saturations than those of Natarajan and Nguyen - and Weber and Newman even though the limiting current densities are comparable. However, without good experimental data on relative permeabilities and the like, one cannot say which approach is more valid. 

The relative permeability is difficult to quantify. It usually is estimated on the basis of laboratory experiments, as a function of relative saturation. In three-phase NAPL-water-air systems, each relative permeability is dependent on the relative saturation of each of the phases. Modeling based on empirical considerations can be employed. For example. Blunt (2000) discusses a model to estimate three-phase relative permeability, based on saturation-weighted interpolation of two-phase relative permeabilities. The model accounts for the trapping of the NAPL and air (gas)... 

Berkowitz B, Emmanuel S, Scher H (2008) Non-Fickian transport and multiple rate mass transfer in porous media Water Resour Res 44, D01 10.1029/2007WR005906 Bijeljic B, Blunt MJ (2006) Pore-scale modeling and continuous time random walk analysis of dispersion in porous media. Water Resour Res 42, W01202, D01 10.1029/2005WR004578 Blunt MJ (2000) An empirical model for three-phase relative permeability. SPE Journal 5 435-445... 

Tortusity factors have been calculated based on different models adopted by different theories. The ribbon and disk-like shapes of nanoclay have been considered in this study. Maji et al. [201] and Sun et al. [202] tabulated most of theories for permeability model equations. The brief descriptions of the theories are tabulated in Table 8. Figure 32 shows the experimental value of relative... 

The steady-state flow numerical experiment was primarily designed to evaluate the phasic relative permeability relations. The numerical experiment is devised within the two-phase lattice Boltzmann modeling framework for the reconstructed CL microstructure, generated using the stochastic reconstruction technique described earlier. Briefly, in the steady-state flow experiment two immiscible fluids are allowed to flow simultaneously until equilibrium is attained and the corresponding saturations, fluid flow rates and pressure gradients can be directly measured and correlated using Darcy s law, defined below. 

Once steady state is achieved, the flux of each phase is calculated. Steady state is considered attained when the saturation and flow rates of both phases do not change any more. The corresponding absolute flux is calculated by modifying the two-phase LB model, where a body force is applied to one phase and the density of the other phase is rendered zero at all locations. Finally, the ratio of flux of each phase from the two-phase calculation to the one obtained from the singlephase calculation gives the relative permeability related to the saturation level. 

Abstract In this contribution, the coupled flow of liquids and gases in capillary thermoelastic porous materials is investigated by using a continuum mechanical model based on the Theory of Porous Media. The movement of the phases is influenced by the capillarity forces, the relative permeability, the temperature and the given boundary conditions. In the examined porous body, the capillary effect is caused by the intermolecular forces of cohesion and adhesion of the constituents involved. The treatment of the capillary problem, based on thermomechanical investigations, yields the result that the capillarity force is a volume interaction force. Moreover, the friction interaction forces caused by the motion of the constituents are included in the mechanical model. The relative permeability depends on the saturation of the porous body which is considered in the mechanical model. In order to describe the thermo-elastic behaviour, the balance equation of energy for the mixture must be taken into account. The aim of this investigation is to provide with a numerical simulation of the behavior of liquid and gas phases in a thermo-elastic porous body. 

A further difficulty that faces oil field modelers is the lack of information they have about downhole conditions and reservoir characteristics. Forward predictions about production are distressingly uncertain. It is therefore common to fit observed production data retrospectively—a procedure known as history matching—and to infer reservoir parameters, particularly permeabilities and relative permeabilities. 

The flow of liquid in ceramic green bodies of liquid volvune fraction, < i[=l - 4>], has been shown to follow permeability models for the flow of liquid relative to ceramic particles [8,9] given in spherical coordi-... 

Macroscopic experiments such as core flooding have been used to obtain relative permeabilities, dispersion coefficients, and other variables relevant to reservoir flow. However, they cannot reveal details of how immiscible phases interact on the pore level. Instead visual experiments have been used to elucidate microscopic flow mechanisms. The latter approach is taken here with experiments using a novel flow cell and state-of-the-art video equipment. The pore level phenomena observed provide a basis for the proper modeling of two-phase flow through porous media at high capillary numbers. 

A comprehensive review of the important factors that affect the flow of emulsions in porous media is presented with particular emphasis on petroleum emulsions. The nature, characteristics, and properties of porous media are discussed. Darcy s law for the flow of a single fluid through a homogeneous porous medium is introduced and then extended for multiphase flow. The concepts of relative permeability and wettability and their influence on fluid flow are discussed. The flow of oil-in-water (OfW) and water-in-oil (W/O) emulsions in porous media and the mechanisms involved are presented. The effects of emulsion characteristics, porous medium characteristics, and the flow velocity are examined. Finally, the mathematical models of emulsion flow in porous media are also reviewed. 

In more realistic situations there is a certain probability of the emulsion droplets coalescing with the bulk oil phase or a part of the bulk oil becoming emulsified. The physics of such complex fiow conditions is not well understood at present. The starting point of describing such a fiow would be to treat it as a normal two-phase flow and use the concept of relative permeability and a model for the rheological properties of the emulsion phase. To account for the material exchange between the bulk phase and the emulsion phase, some form of droplet population balance model will be needed. 

In the original Buckley-Leverett theory, gravitational, compressibility and capillarity are ignored. Devereux (36) presents the solution for the case of constant pressure, and the constant-velocity case was derived by Soo and Radke 12). The model requires a knowledge of the capillary retarding force per unit volume of the porous medium, and the relative permeabilities of the oil droplets in the emulsion and the continuous water phase. These relative permeabilities are assumed to be functions of the oil saturation in the porous medium. These must be determined before the model can be used. 

Gas relative permeability, Pk, is defined as the permeability of a fluid through a porous medium partially blocked by a second fluid, normalized by the permeability when the pore space is free of this second fluid. This property diminishes at the percolation threshold , at which a significant portion of the pores are still conducting but they do not form a continuous path along the flow direction. It is obvious that only the network model, can provide a satisfactory analysis of the percolation threshold problem. Nicholson et al. [3] introduced a simple network model, and applied it on gas relative permeability [4]. For the gas relative permeability, an explicit approximate analytical relation between the relative permeability and the two network parameters, namely z and the first four moments of, f(r), has been developed, based on the Effective Medium Approximation (EMA) [5]. If a porous... 

Figure 2a. Relative permeability vs adsorbate Figure 2b. Relative permeability vs open volume, for 3D network and EMA models pores fraction, fh, for 3D network and EMA...

In Fig. 2 relative permeability curves computed by the network model are plotted for r=4,6 and 8 and are compared with EMA results. As r increases the Pr curve becomes broader as it approaches the percolation threshold, Vsr- In all cases EMA is in good agreement with the network solution, except in the neighborhood of Kv ... 

Since the dominant feature of packings of spheroidal particles is the constrictions between the tetrahedral cavities formed by the Alumina microspheres, a more realistic model is required, based on the random sphere packing models. Such models are obviously more complex. Conversely, they permit a more realistic representation of the pore space among the spheroidal particles. A preliminary model has been reported for sorption [20] and relative permeability Pr [21]. 

Scandellari Nilsen, L., 0ren, P.E., Bakke, S. and Henriquez, A. 1996. Prediction of relative permeability and capillary pressure from a pore model. SPE 35531. 

For many years, the rate and extent of absorption in the small intestine were thought to be determined solely by the lipid/water solubility and membrane permeability characteristics of the drug. While this relatively simplistic model worked for many drugs, there are a number of exceptions to this rule, suggesting other forces are at work within the GI system to control the absorption of drugs. It is now known that a complex system of transporter proteins and metabolic enzymes is present within the GI system. Expression of influx transporters in the intestinal epithelial cells can increase absorption of drugs that are substrates for these transporters, whereas efflux transporters can reduce oral absorption of these drugs. In particular, the impact of the... 

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